Connection to the Truchet Tile System

This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ‎(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.